data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "How many 3-digit numbers have the property that the units digit is at least twice the tens digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"split": "train",
"index": 900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of 10 numbers is 85. If the numbers 70 and 76 are removed from the set of numbers, what is the average of the remaining numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a s... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the area of a circle is less than $60\\pi$ square inches, what is the greatest possible integer value in inches of the radius of the circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\\times11$ rectangle. How many red squares does it contain? (The sides of all considered re... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 905,
"solution": "1. **Identify the pattern and constraints**: The students are arranged in different configurations each day, with a unique number of students per row for 12 consecutive days. On the 13th day, no new configuration is possible. This implies that the total number of stu... |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\triangle ABE$, $\\triangle BCE$, and $\\triangle CDE$ are right-angled with $\\angle AEB = 30^\\circ, \\angle BEC = 45^\\circ$, and $\\angle CED = 45^\\circ$, and $AE=30$. Find the length of $CE$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Santa Claus has a sack containing both chocolate and gummy candies, totaling 2023 pieces. The chocolate candies make up 75% of the gummy candies. How many chocolate candies does Santa Claus have in his sack? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "867"
} | {
"split": "train",
"index": 907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the range of the function $f(x) = \\arcsin x + \\arccos x + \\arctan x.$ All functions are in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left[ \\frac{\\pi}{4}, \\frac{3 \\pi}{4} \\right]"
} | {
"split": "train",
"index": 908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a right triangle is rotated about one leg, the volume of the cone produced is $800\\pi \\;\\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\\pi \\;\\textrm{ cm}^3$. What is the length (in cm) of the hypotenuse of t... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 909,
"solution": "Let $a$ and $b$ be the two legs of the equation. We can find $\\frac{a}{b}$ by doing $\\frac{1920\\pi}{800\\pi}$. This simplified is $\\frac{12}{5}$. We can represent the two legs as $12x$ and $5x$ for $a$ and $b$ respectively.\n Since the volume of the first cone is... |
deepscaler | [
{
"role": "user",
"content": "If $\\sqrt{25-\\sqrt{n}}=3$, what is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 910,
"solution": "Since $\\sqrt{25-\\sqrt{n}}=3$, then $25-\\sqrt{n}=9$. Thus, $\\sqrt{n}=16$ and so $n=16^{2}=256$."
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)$ ($x \\in \\mathbb{R}$) that satisfies $f(x+\\pi)=f(x)+\\sin x$, and $f(x)=0$ when $0 \\leqslant x < \\pi$, determine the value of $f(\\frac{23\\pi}{6})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\\triangle ABC$, respectively, and $\\overrightarrow{m}=(a,−\\sqrt {3}b)$, $\\overrightarrow{n}=(\\sin B,\\cos A)$, if $a= \\sqrt {7}$, $b=2$, and $\\overrightarrow{m} \\perp \\overrightar... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the base five product of the numbers $132_{5}$ and $12_{5}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2114_5"
} | {
"split": "train",
"index": 913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n$ such that: \n\\[ \\dfrac{n^3+3}{n^2+7} \\]\nis a positive integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\text{ and } 5"
} | {
"split": "train",
"index": 914,
"solution": "To determine the positive integers \\( n \\) such that the expression \n\n\\[\n\\dfrac{n^3 + 3}{n^2 + 7}\n\\]\n\nis a positive integer, we need to analyze when the expression simplifies to a whole number.\n\n### Step 1: Dividing the Polynomials\n\nConsider the divisi... |
deepscaler | [
{
"role": "user",
"content": "Two circles \\( C_{1} \\) and \\( C_{2} \\) have their centers at the point \\( (3, 4) \\) and touch a third circle, \\( C_{3} \\). The center of \\( C_{3} \\) is at the point \\( (0, 0) \\) and its radius is 2. What is the sum of the radii of the two circles \\( C_{1} \\) and ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $bc=3$, $a=2$, then the minimum value of the area of the circumcircle of $\\triangle ABC$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi}{8}"
} | {
"split": "train",
"index": 916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the four-digit numbers, the number of four-digit numbers that have exactly 2 digits repeated is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3888"
} | {
"split": "train",
"index": 917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan\\left( \\frac{\\pi}{4} + \\alpha \\right) = \\frac{1}{7}$, with $\\alpha \\in \\left( \\frac{\\pi}{2}, \\pi \\right)$, find the value of $\\tan\\alpha$ and $\\cos\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{5}"
} | {
"split": "train",
"index": 918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Isosceles triangle $ABE$ of area 100 square inches is cut by $\\overline{CD}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 75 square inches. If the altitude of triangle $ABE$ from $A$ is 20 inches, what is the number of inches in th... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of $ \\int_0^1 |e^{ \\minus{} x} \\minus{} a|dx\\ ( \\minus{} \\infty < a < \\infty)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - 2e^{-1}"
} | {
"split": "train",
"index": 920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given three real numbers \\(p, q,\\) and \\(r\\) such that\n\n\\[ p+q+r=5 \\quad \\text{and} \\quad \\frac{1}{p+q}+\\frac{1}{q+r}+\\frac{1}{p+r}=9 \\]\n\nWhat is the value of the expression\n\n\\[ \\frac{r}{p+q}+\\frac{p}{q+r}+\\frac{q}{p+r} ? \\] Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the integral \\(\\int_{0}^{\\pi / 2} \\frac{\\sin^{3} x}{2 + \\cos x} \\, dx\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\ln \\left(\\frac{2}{3}\\right) + \\frac{3}{2}"
} | {
"split": "train",
"index": 922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(m,1)$ and $\\overrightarrow{b}=(4-n,2)$, with $m > 0$ and $n > 0$. If $\\overrightarrow{a}$ is parallel to $\\overrightarrow{b}$, find the minimum value of $\\frac{1}{m}+\\frac{8}{n}$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "They paid 100 rubles for a book and still need to pay as much as they would need to pay if they had paid as much as they still need to pay. How much does the book cost? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$. If the area of triangle $ABE$ is 50 square units, and the area of triangle $ADE$ is 20 square units, what is the area of trapezoid $ABCD$? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and\n\\[f(x - y) = \\sqrt{f(xy) + 2}\\]for all $x > y > 0.$ Determine $f(2009).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( x \\) is a real number and \\( \\lceil x \\rceil = 14 \\), how many possible values are there for \\( \\lceil x^2 \\rceil \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number $r$ with the following property: For every choice of $2023$ unit vectors $v_1,v_2, \\dots ,v_{2023} \\in \\mathbb{R}^2$ , a point $p$ can be found in the plane such that for each subset $S$ of $\\{1,2, \\dots , 2023\\}$ , the su... | math | {
"style": "rule",
"ground_truth": "\\frac{2023}{2}"
} | {
"split": "train",
"index": 928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(a^{2} - 1 = 123 \\times 125\\) and \\(a > 0\\), find \\(a\\).\n\nIf the remainder of \\(x^{3} - 16x^{2} - 9x + a\\) when divided by \\(x - 2\\) is \\(b\\), find \\(b\\).\n\nIf an \\(n\\)-sided polygon has \\((b + 4)\\) diagonals, find \\(n\\).\n\nIf the points \\((3, n), ... | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\(b\\) such that the graph of the equation \\[ 3x^2 + 9y^2 - 12x + 27y = b\\] represents a non-degenerate ellipse? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{129}{4}"
} | {
"split": "train",
"index": 930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Professor Lee has ten different mathematics books on a shelf, consisting of three calculus books, four algebra books, and three statistics books, determine the number of ways to arrange the ten books on the shelf keeping all calculus books together and all statistics book... | math | {
"style": "rule",
"ground_truth": "25920"
} | {
"split": "train",
"index": 931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers \\( x \\) and \\( y \\) satisfy the equation \\( 280x^{2} - 61xy + 3y^{2} - 13 = 0 \\) and are respectively the fourth and ninth terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} x-3 & \\quad \\text{ if } x < 5 \\\\ \\sqrt{x} & \\quad \\text{ if } x \\ge 5 \\end{aligned} \\right.\\]has an inverse $f^{-1}.$ Find the value of $f^{-1}(-6) + f^{-1}(-5) + \\dots + f^{-1}(5) + f^{-1}(6).$ Let's think step by s... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"split": "train",
"index": 933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is divided into 2016 triangles, with no vertex of any triangle lying on the sides or inside any other triangle. The sides of the square are sides of some of the triangles in the division. How many total points, which are the vertices of the triangles, are located insid... | math | {
"style": "rule",
"ground_truth": "1007"
} | {
"split": "train",
"index": 934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of the binomial ${(\\sqrt{x}-\\frac{1}{{2x}}})^n$, only the coefficient of the 4th term is the largest. The constant term in the expansion is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{4}"
} | {
"split": "train",
"index": 935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anthony made $5$ of his first $12$ free throw attempts. If he makes $2/3$ of his next $24$ attempts, by how many percentage points will he increase his overall success rate percentage? Express your answer to the nearest whole number. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "17\\%"
} | {
"split": "train",
"index": 936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\theta$ be the angle between the planes $2x + y - 2z + 3 = 0$ and $6x + 3y + 2z - 5 = 0.$ Find $\\cos \\theta.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{21}"
} | {
"split": "train",
"index": 937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ms. Carr asks her students to read any $5$ of the $10$ books on a reading list. Harold randomly selects $5$ books from this list, and Betty does the same. What is the probability that there are exactly $2$ books that they both select? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{63}"
} | {
"split": "train",
"index": 938,
"solution": "1. **Calculate the total number of ways Harold and Betty can each choose 5 books from 10 books:**\n The number of ways to choose 5 books from 10 is given by the binomial coefficient $\\binom{10}{5}$. Since both Harold and Betty are choosing 5 books independently, t... |
deepscaler | [
{
"role": "user",
"content": "Little Wang has three ballpoint pens of the same style but different colors. Each pen has a cap that matches its color. Normally, Wang keeps the pen and cap of the same color together, but sometimes he mixes and matches the pens and caps. If Wang randomly pairs the pens and cap... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a group of $3$ orthopedic surgeons, $4$ neurosurgeons, and $5$ internists, a medical disaster relief team of $5$ people is to be formed. How many different ways can the team be selected such that there is at least one person from each specialty? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "590"
} | {
"split": "train",
"index": 940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equation of curve $C\\_1$ as $\\begin{cases} x=2\\cos \\theta \\ y=\\sqrt{3}\\sin \\theta \\end{cases}(\\theta \\text{ is the parameter})$, and curve $C\\_2$ has a polar coordinate equation of $\\rho=2$ with the origin $O$ as the pole and the positive semi-... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"split": "train",
"index": 941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the populations of three communities $A$, $B$, and $C$ are 600, 1200, and 1500 residents respectively, and if 15 people are drawn from community $C$, determine the total number of people drawn in the sample. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all angles $\\theta,$ $0 \\le \\theta \\le 2 \\pi,$ with the following property: For all real numbers $x,$ $0 \\le x \\le 1,$\n\\[x^2 \\cos \\theta - x(1 - x) + (1 - x)^2 \\sin \\theta > 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{\\pi}{12}, \\frac{5 \\pi}{12} \\right)"
} | {
"split": "train",
"index": 943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer less than or equal to \\[\\frac{5^{105} + 4^{105}}{5^{99} + 4^{99}}?\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15624"
} | {
"split": "train",
"index": 944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 945,
"solution": "1. **Expression Analysis**:\n We start by analyzing the polynomial \\( P = x^4 + 6x^3 + 11x^2 + 3x + 31 \\). We aim to express \\( P \\) in a form that involves a square of an expression, as we are looking for \\( P \\) to be a perfect square.\n\n2. **Initial Facto... |
deepscaler | [
{
"role": "user",
"content": "Write the number 2013 several times in a row so that the resulting number is divisible by 9. Explain the answer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "201320132013"
} | {
"split": "train",
"index": 946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the coefficients in the polynomial $-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that\n\\[\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = 6.\\]Find the minimum value of $x^3 y^2 z.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{108}"
} | {
"split": "train",
"index": 948,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the following expression:\n\n$$5x + 6 - x + 12$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4x + 18"
} | {
"split": "train",
"index": 949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Parallelogram $ABCD$ has vertices $A(3,3)$, $B(-3,-3)$, $C(-9,-3)$, and $D(-3,3)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not above the $x$-axis? Express your answer as a common fraction. Let's... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find an axis of symmetry for the function $f(x) = \\cos(2x + \\frac{\\pi}{6})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"split": "train",
"index": 951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given four points \\( A, B, C, \\) and \\( D \\) in space that are not coplanar, each pair of points is connected by an edge with a probability of \\( \\frac{1}{2} \\). The events of whether any two pairs of points are connected are mutually independent. What is the probability... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the second quadrant, and $\\tan(\\theta + \\frac{\\pi}{4}) = \\frac{1}{2}$, calculate the value of $\\sin\\theta + \\cos\\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{10}}{5}"
} | {
"split": "train",
"index": 953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation \\(x + 11y + 11z = n\\) where \\(n \\in \\mathbf{Z}_{+}\\), there are 16,653 sets of positive integer solutions \\((x, y, z)\\). Find the minimum value of \\(n\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"split": "train",
"index": 954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x \\neq 0$, $\\frac{x}{2} = y^2$ and $\\frac{x}{4} = 4y$, then $x$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 955,
"solution": "1. **Start by expressing $x$ in terms of $y$ from the second equation:**\n \\[\n \\frac{x}{4} = 4y \\implies x = 16y\n \\]\n\n2. **Substitute $x = 16y$ into the first equation:**\n \\[\n \\frac{x}{2} = y^2 \\implies \\frac{16y}{2} = y^2 \\implies 8y = y^2\n... |
deepscaler | [
{
"role": "user",
"content": "For $i = 1, 2, \\cdots, n$, if $\\left|x_{i}\\right| < 1$ and $\\left|x_{1}\\right| + \\left|x_{2}\\right| + \\cdots + \\left|x_{n}\\right| = 2005 + \\left|x_{1} + x_{2} + \\cdots + x_{n} \\right|$, find the minimum value of the positive integer $n$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"split": "train",
"index": 956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To arrange a class schedule for one day with the subjects Chinese, Mathematics, Politics, English, Physical Education, and Art, where Mathematics must be in the morning and Physical Education in the afternoon, determine the total number of different arrangements. Let's think st... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be a positive integer and let the lines $13x+11y=700$ and $y=mx-1$ intersect in a point whose coordinates are integers. Then m can be: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 958,
"solution": "1. **Substitute the equation of the line $y = mx - 1$ into the equation $13x + 11y = 700$:**\n \\[\n 13x + 11(mx - 1) = 700\n \\]\n Simplifying this, we get:\n \\[\n 13x + 11mx - 11 = 700\n \\]\n \\[\n (13 + 11m)x = 711\n \\]\n\n2. **Factorize 711... |
deepscaler | [
{
"role": "user",
"content": "The intercept on the x-axis, the intercept on the y-axis, and the slope of the line 4x-5y-20=0 are respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{5}"
} | {
"split": "train",
"index": 959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When drawing a histogram of the lifespans of 1000 people, if the class interval is uniformly 20, and the height of the vertical axis for the age range 60 to 80 years is 0.03, calculate the number of people aged 60 to 80. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $c$ if the roots of the quadratic $9x^2 - 5x + c$ are $\\frac{-5\\pm i\\sqrt{415}}{18}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{110}{9}"
} | {
"split": "train",
"index": 961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gracie and Joe are choosing numbers on the complex plane. Joe chooses the point $1+2i$. Gracie chooses $-1+i$. How far apart are Gracie and Joe's points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\triangle PQR$ is similar to $\\triangle STU$. The length of $\\overline{PQ}$ is 10 cm, $\\overline{QR}$ is 12 cm, and the length of $\\overline{ST}$ is 5 cm. Determine the length of $\\overline{TU}$ and the perimeter of $\\triangle STU$. Express your answer as a decimal. Let... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$x$ is a real number with the property that $x+\\tfrac1x = 3$. Let $S_m = x^m + \\tfrac{1}{x^m}$. Determine the value of $S_7$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "843"
} | {
"split": "train",
"index": 964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some squares of a $n \\times n$ table $(n>2)$ are black, the rest are white. In every white square we write the number of all the black squares having at least one common vertex with it. Find the maximum possible sum of all these numbers. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "3n^{2}-5n+2"
} | {
"split": "train",
"index": 965,
"solution": "The answer is $3n^{2}-5n+2$. The sum attains this value when all squares in even rows are black and the rest are white. It remains to prove that this is the maximum value. The sum in question is the number of pairs of differently coloured squares sharing at least one... |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\dfrac{5+12i}{2-3i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2+3i"
} | {
"split": "train",
"index": 966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of the geometric sequence $\\{a\\_n\\}$ is $S\\_n$, and it satisfies $S\\_n=(\\frac{1}{2})^{n}-1$, find the limit as $n$ approaches infinity of $(a\\_1+a\\_3+...+a\\_2n-1)$ . Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( \\sqrt{100-x}=9 \\), what is the value of \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 968,
"solution": "Since \\( \\sqrt{100-x}=9 \\), then \\( 100-x=9^{2}=81 \\), and so \\( x=100-81=19 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\\omega$ and bisect its perimet... | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"split": "train",
"index": 969,
"solution": "\nTo approach this problem, we will analyze the geometric properties and symmetrical nature of the heart shape and the properties of the cyclic quadrilateral \\(XYZT\\).\n\n1. **Understanding the Geometry of the Heart Shape:**\n - The heart shape is constructed fro... |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\lg 2 = 0.30103\\), calculate the number of digits in \\( M = 1 + 10^4 + \\frac{10^4 (10^4 - 1)}{1 \\cdot 2} + \\frac{10^4 (10^4-1)(10^4-2)}{1 \\cdot 2 \\cdot 3} + \\cdots + \\frac{10^4 (10^4 - 1)}{1 \\cdot 2} + 10^4 + 1\\). Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "3011"
} | {
"split": "train",
"index": 970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A supermarket has 6 checkout lanes, each with two checkout points numbered 1 and 2. Based on daily traffic, the supermarket plans to select 3 non-adjacent lanes on Monday, with at least one checkout point open in each lane. How many different arrangements are possible for the c... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x \\) and \\( y \\) be positive numbers, and let \\( s \\) be the smallest of the numbers \\( x \\), \\( y + \\frac{1}{x} \\), and \\( \\frac{1}{y} \\). Find the maximum possible value of \\( s \\). For which values of \\( x \\) and \\( y \\) is it achieved? Let's think... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two numbers are randomly chosen without replacement from the set $\\{6, 8, 9, 12\\}$, what is the probability that their product will be a multiple of 36? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A teacher has prepared three problems for the class to solve. In how many different ways can he present these problems to the students if there are 30 students in the class? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24360"
} | {
"split": "train",
"index": 974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integer solutions to $n^{x}+n^{y}=n^{z}$ with $n^{z}<2001$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 975,
"solution": "If $n=1$, the relation can not hold, so assume otherwise. If $x>y$, the left hand side factors as $n^{y}\\left(n^{x-y}+1\\right)$ so $n^{x-y}+1$ is a power of $n$. But it leaves a remainder of 1 when divided by $n$ and is greater than 1, a contradiction. We reach a s... |
deepscaler | [
{
"role": "user",
"content": "A 9 by 9 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice plays a game where she rolls a fair eight-sided die each morning. If Alice rolls a number divisible by 3, she wears red; otherwise, she wears blue. If she rolls a perfect square (1, 4), then she rolls again. In a leap year, what is the expected number of times Alice will ... | math | {
"style": "rule",
"ground_truth": "488"
} | {
"split": "train",
"index": 977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two types of anti-inflammatory drugs must be selected from $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$, with the restriction that $X_{1}$ and $X_{2}$ must be used together, and one type of antipyretic drug must be selected from $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$, with the furth... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pentagon is formed by placing an equilateral triangle on top of a rectangle. The side length of the equilateral triangle is equal to the width of the rectangle, and the height of the rectangle is twice the side length of the triangle. What percent of the area of the pentagon ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{\\sqrt{3} + 8} \\times 100\\%"
} | {
"split": "train",
"index": 979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Edward stopped to rest at a place 1,875 feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of \\( 100 \\, \\mathrm{ft/s} \\). At the same time, Edward started running away with an acceleration of \\( 1 \\, \\mathrm... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{2}}{3}$ ($\\frac{\\pi}{2} < \\alpha < \\pi$), find the values of the following expressions: \n$(1) \\sin \\alpha - \\cos \\alpha$; \n$(2) \\sin^2\\left(\\frac{\\pi}{2} - \\alpha\\right) - \\cos^2\\left(\\frac{\\pi}{2} + \\a... | math | {
"style": "rule",
"ground_truth": "-\\frac{4\\sqrt{2}}{9}"
} | {
"split": "train",
"index": 981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A manager schedules an informal review at a café with two of his team leads. He forgets to communicate a specific time, resulting in all parties arriving randomly between 2:00 and 4:30 p.m. The manager will wait for both team leads, but only if at least one has arrived before h... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If set $A=\\{x\\in N\\left|\\right.-1 \\lt x\\leqslant 2\\}$, $B=\\{x\\left|\\right.x=ab,a,b\\in A\\}$, then the number of non-empty proper subsets of set $B$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\sqrt{(3-2\\sqrt{3})^2}+\\sqrt{(3+2\\sqrt{3})^2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 984,
"solution": "1. **Define the expression and simplify each square root:**\n Let $x = \\sqrt{(3-2\\sqrt{3})^2} + \\sqrt{(3+2\\sqrt{3})^2}$.\n\n Since squaring a real number and then taking the square root gives the absolute value of the original number, we have:\n \\[\n \\s... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\begin{cases} \\left( \\frac {1}{2}\\right)^{x} & (x\\geqslant 4) \\\\ f(x+1) & (x < 4)\\end{cases}$, find the value of $f(\\log _{2}3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"split": "train",
"index": 985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{1}{2},\\frac{2}{3}\\right)"
} | {
"split": "train",
"index": 986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integers $(x, y)$ with $x, y \\leq 2020$ such that $3 x^{2}+10 x y+3 y^{2}$ is the power of some prime. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 987,
"solution": "We can factor as $(3 x+y)(x+3 y)$. If $x \\geq y$, we need $\\frac{3 x+y}{x+3 y} \\in\\{1,2\\}$ to be an integer. So we get the case where $x=y$, in which we need both to be a power of 2, or the case $x=5 y$, in which case we need $y$ to be a power of 2. This gives u... |
deepscaler | [
{
"role": "user",
"content": "Consider the equation $p = 15q^2 - 5$. Determine the value of $q$ when $p = 40$.\nA) $q = 1$\nB) $q = 2$\nC) $q = \\sqrt{3}$\nD) $q = \\sqrt{6}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "q = \\sqrt{3}"
} | {
"split": "train",
"index": 988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$ . Let $A_0, A_1, A_2,\\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such tha... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T = \\{9^k : k ~ \\mbox{is an integer}, 0 \\le k \\le 4000\\}$. Given that $9^{4000}_{}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T_{}^{}$ have 9 as their leftmost digit? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "184"
} | {
"split": "train",
"index": 990,
"solution": "Since $9^{4000}$ has 3816 digits more than $9^1$, $4000 - 3816 = \\boxed{184}$ numbers have 9 as their leftmost digits."
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$ . A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\\gcd (c,d) = 70$ and $\\operatorname{lcm} (c,d) = 630{,}000$ is $\... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\), let \\( E \\) be the point where the side \\( AC \\) is divided into quarters closest to \\( C \\), and let \\( F \\) be the midpoint of side \\( BC \\). The line passing through points \\( E \\) and \\( F \\) intersects line \\( AB \\) at point \\( D \... | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"split": "train",
"index": 992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A weightless pulley has a rope with masses of 3 kg and 6 kg. Neglecting friction, find the force exerted by the pulley on its axis. Consider the acceleration due to gravity to be $10 \\, \\mathrm{m/s}^2$. Give the answer in newtons, rounding it to the nearest whole number if ne... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the height of a cylinder is $1$, and the circumferences of its two bases are on the surface of the same sphere with a diameter of $2$, calculate the volume of the cylinder. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\pi}{4}"
} | {
"split": "train",
"index": 994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Manya has a stack of $85=1+4+16+64$ blocks comprised of 4 layers (the $k$ th layer from the top has $4^{k-1}$ blocks). Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only bl... | math | {
"style": "rule",
"ground_truth": "3384"
} | {
"split": "train",
"index": 995,
"solution": "Each time Laura removes a block, 4 additional blocks are exposed, increasing the total number of exposed blocks by 3 . She removes 5 blocks, for a total of $1 \\cdot 4 \\cdot 7 \\cdot 10 \\cdot 13$ ways. However, the stack originally only has 4 layers, so we must sub... |
deepscaler | [
{
"role": "user",
"content": "\nChicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, a... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the following quadratic equation: $x^2 + 5x - 4 = 0.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-5 - \\sqrt{41}}{2}"
} | {
"split": "train",
"index": 997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $n$ satisfy\\[\\dfrac{n+1000}{70} = \\lfloor \\sqrt{n} \\rfloor?\\](Recall that $\\lfloor x\\rfloor$ is the greatest integer not exceeding $x$.)\n$\\textbf{(A) } 2 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 6 \\qquad\\textbf{(D) } 30 \\qquad\\textb... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Come up with at least one three-digit number PAU (all digits are different), such that \\((P + A + U) \\times P \\times A \\times U = 300\\). (Providing one example is sufficient) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "235"
} | {
"split": "train",
"index": 999,
"solution": ""
} |
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